Statistical theory of shot noise in quasi-one-dimensional field-effect

PHYSICAL REVIEW B 81, 035329 共2010兲

Statistical theory of shot noise in quasi-one-dimensional field-effect transistors in the presence of electron-electron interaction Alessandro Betti, Gianluca Fiori, and Giuseppe Iannaccone Dipartimento di Ingegneria dell’Informazione: Elettronica, Informatica, Telecomunicazioni, Università di Pisa, Via Caruso 16, 56122 Pisa, Italy 共Received 27 April 2009; revised manuscript received 19 October 2009; published 21 January 2010兲 We present an expression for the shot noise power spectral density in quasi-one dimensional conductors electrostatically controlled by a gate electrode, which includes the effects of Coulomb interaction and of Pauli exclusion among charge carriers. In this sense, our expression extends the well known Landauer-Büttiker noise formula to include the effect of Coulomb interaction inducing fluctuations of the potential in the device region. Our approach is based on evaluating the statistical properties of the scattering matrix and on a secondquantization many-body description. From a quantitative point of view, statistical properties are obtained by means of Monte Carlo simulations on an ensemble of different configurations of injected states, requiring the solution of the Poisson-Schrödinger equation on a three-dimensional grid, with the nonequilibrium Green’s functions formalism. In a series of examples, we show that failure to consider the effects of Coulomb interaction on noise leads to a gross overestimation of the noise spectrum of quasi-one-dimensional devices. DOI: 10.1103/PhysRevB.81.035329

PACS number共s兲: 73.50.Td, 73.63.Nm

I. INTRODUCTION

As quasi-one-dimensional field-effect transistors 共FETs兲, based for example on Carbon nanotubes 共CNTs兲 or Silicon nanowires 共SNWs兲, are increasingly investigated as a possible replacement for conventional planar FETs, it is important to achieve complete understanding of the properties of shot noise of one-dimensional conductors electrostatically controlled by a third 共gate兲 electrode. Shot noise is particularly sensitive to carrier-carrier interaction, which in turn can be particularly significant in one-dimensional nanoscale conductors, where electrons are few and screening is limited.1 Low-frequency 1 / f noise in quasi one-dimensional conductors has been the subject of interest for several authors,2–4 whereas few experimental papers on shot noise have recently been published.5,6 Due to the small amount of mobile charge in nanoscale one-dimensional FETs, even in strong inversion, drain current fluctuations can heavily affect device electrical behavior. Of course, noise is an unavoidable and undesirable feature of electron devices, and its effect must be minimized or kept within tolerable levels for the operation of electronic circuits. From a more fundamental point of view, it is also a rich source of information on electron-electron interaction, which cannot be obtained from dc or ac electrical characteristics. The main sources of noise are injection from the contacts into the device region, through the random occupation of states around the Fermi energy at the contacts, and partial transmission of electrons through the conductor, which gives rise to the so-called partition noise. The main types of interaction that have a clear effect on noise are Pauli exclusion, which reduces fluctuations of the rate of injected electrons by limiting the occupancy of injected states, and Coulomb repulsions among electrons, which is the cause of fluctuations of the potential in the device region, that often suppress, but sometimes enhance the effect of fluctuations in the rate of injected electrons. The combined effect of Pauli exclusion and Coulomb repulsion on shot noise has been investigated in the case of 1098-0121/2010/81共3兲/035329共10兲

ballistic double gate MOSFETs,7 in planar MOSFETs8 and in resonant tunneling diodes.9–11 There are still few attempts12 to a complete quantitative understanding of shot noise in ballistic CNT and SNW-FETs. Indeed, when addressing a resonant tunneling diode one can usually adopt an approach that exploits the fact that the two opaque barriers break the device in three loosely coupled regions 共the two contacts and the well兲, among which transitions can described by Fermi golden rule, as has been done in Refs. 9–11. This is not possible in the case of a transistor, where coupling between the channel and the contacts is very good. Another important issue is represented by the fact that the widely known Landauer-Büttiker’s noise formula,13,14 does not take into account the effect of Coulomb interaction on shot noise through potential fluctuations. Indeed, recent experiments on shot noise in CNT-based Fabry Perot interferometers6 show that in some bias conditions manybody corrections might be needed to explain the observed noise suppression. Other experiments show that at low temperature suspended ropes of single-wall carbon nanotubes of length 0.4 ␮m exhibit a significant suppression of current fluctuations by a factor smaller than 1/100 compared to full shot noise.5 However, this experimental result is not supported by a convincing interpretation, since possible explanations extend from ballistic transport in a small number of tubes within a rope, to diffusive transport in a substantial fraction of the CNTs. In this work, we present an expression for the shot noise power spectral density of ballistic quasi-one-dimensional channels based on a statistical approach relying on quantities obtained from Monte Carlo 共MC兲 simulations of randomly injected electrons from the reservoirs. The expression is derived within the second quantization formalism, and simulations are based on the self-consistent 共SC兲 solution of the three-dimensional 共3D兲 Poisson and Schrödinger equations, within the nonequilibrium Green’s function 共NEGF兲 formalism.15 Our proposed expression generalizes the LandauerBüttiker’s noise formula including the effects of Coulomb

035329-1

©2010 The American Physical Society


BETTI, FIORI, AND IANNACCONE

bSm

interaction, which is significant for a large class of devices, and in particular for one-dimensional conductors.

b Dn

channel

SOURCE

DRAIN

II. THEORY

According to Milatz’s theorem,16 the power spectral density of the noise current in the zero frequency limit can be written as S共0兲 = lim1/␯→⬁关2 / ␯ · var共I兲兴, where ␯ is the injection rate of a carrier from a contact and var共I兲 is the variance of the current. According to Ref. 17, ␯ can be expressed as ␯ = ⌬E / 共2␲ប兲 where ⌬E is the energy discretization step, i.e., the minimum energy separation between injected states. Indeed, the contribution to the current of a transverse mode in the energy interval ⌬E can be expressed in the zerotemperature limit by the Landauer-Büttiker formula as 具dI典 = e / 共2␲ប兲⌬E. On the other hand 具dI典 = e␯, from which ␯ = ⌬E / 共2␲ប兲 derives. Finally, the power spectral density of shot noise at zero frequency can be expressed as 2 var共I兲 . S共0兲 = lim var共I兲 = lim 4␲ប ⌬E ␯→0 ␯ ⌬E→0

共1兲

The variance of the current can be derived by means of the second quantization formalism, which allows a concise treatment of the many-electron problem. Let us consider a mesoscopic conductor connected to two reservoirs 关source 共S兲 and drain 共D兲兴, where electron states are populated according to their Fermi occupation factors 共Fig. 1兲. For simplicity, we assume that the conductor is sufficiently short as to completely neglect inelastic scattering events. Thermalization occurs only in the reservoirs. At zero magnetic field and far from the interacting channel, the timedependent current operator at the source can be expressed as the difference between the occupation number of carriers + − 兲 and outward 共NSm 兲 in each quantum moving inward 共NSm 13 channel m, I共t兲 =

e 兺 h m苸S

冕

+ − dE关NSm 共E,t兲 − NSm 共E,t兲兴,

共2兲

where + NSm 共E,t兲 =

− 共E,t兲 = NSm

冕

+ d共ប␻兲aSm 共E兲aSm共E + ប␻兲e−i␻t ,

冕

+ d共ប␻兲bSm 共E兲bSm共E + ប␻兲e−i␻t .

共3兲

† The introduced operators aSm 共E兲 and aSm共E兲 create and annihilate, respectively, incident electrons in the source lead with total energy E in the channel m 共Fig. 1兲. In the same † 共E兲 and annihilation bSm共E兲 operators way, the creation bSm refer to electrons in the source contact for outgoing states. The channel index m runs over all the transverse modes and different spin orientations. The operators a and b are related via a unitary transformation 共n = 1 , . . . , WS兲,13

a

aSm

Dn

n=1,...,WD

m=1,..., WS

FIG. 1. Annihilation operators for ingoing 共aSm , aDn兲 and outgoing electron states 共bSm , bDn兲 in a two terminal scattering problem 共m = 1 , . . . , WS ; n = 1 , . . . , WD兲. WS

bSn共E兲 =

兺

WD

rnm共E兲aSm共E兲 +

m=1

⬘ 共E兲aDm共E兲, 兺 tnm

共4兲

m=1

where WS and WD represent the number of quantum channels in the source and drain leads, respectively, while the blocks r 共size WS ⫻ WS兲 and t⬘ 共size WS ⫻ WD兲, describe electron reflection at the source 共r兲 and transmission from drain to source 共t⬘兲 and are included in the scattering matrix s as18 s=

冉冊

r t⬘ . t r⬘

共5兲

The dimensions of s are 共WS + WD兲 ⫻ 共WS + WD兲. Blocks t and r⬘ in Eq. 共5兲 are related to source-to-drain transmission and reflection back to the drain, respectively. In the following, time dependence will be neglected, since we are interested to the zero frequency case. If we denote with 兩␴典 a many-particle 共antisymmetrical兲 state, the occupation number in the reservoir ␣ in the channel m can be expressed as ␴␣m共E兲 = 具a␣† m共E兲a␣m共E兲典␴. Pauli exclusion principle does not allow two electrons to occupy the same spin orbital, therefore ␴␣m共E兲 can be either 0 or 1. In addition, since fluctuations of the potential profile along the channel due to Coulomb interaction between randomly injected carriers affect the transmission of electrons, the scattering matrix elements have to depend on the occupation numbers of all states in both reservoirs: s共E兲 = s关␴S1共E兲 , ␴S2共E兲 , ¯ , ␴D1共E兲 , ␴D2共E兲 , ¯兴. Let us stress the fact that, as pointed out in Ref. 13, whenever a finite channel is connected to semi-infinite leads, the channel can be considered as a small perturbation to the equilibrium regime of the contacts, and independent random statistics can be used for both reservoirs. According to Ref. 13, current fluctuations can be evaluated by introducing an ensemble of many electrons states 兵兩␴1典 , 兩␴2典 , 兩␴3典 , ¯ , 兩␴N典其 and by weighting each state properly, i.e., by finding its statistical average, denoted by 具典s. Each reservoir ␣共␣ = S , D兲 is assumed to be at thermal equilibrium, so that its average occupancy can be described by the Fermi-Dirac statistics f ␣. As a consequence, the statistical average of ␴␣m共E兲 reads,13 具␴␣m共E兲典s = 具具a␣† m共E兲a␣m共E兲典␴典s = f ␣共E兲.

共6兲

Neglecting correlations between the occupation numbers of the same quantum channel at different energies, or between

035329-2


STATISTICAL THEORY OF SHOT NOISE IN QUASI-…

different channels at the same energy, we obtain,13

具I2典 =

具␴␣m共E兲␴␤n共E⬘兲典s = f ␣共E兲f ␤共E⬘兲,

共7兲

= f ␣共E兲f ␤共E⬘兲 + ␦共E − E⬘兲␦␣␤␦mn关f ␣共E兲 − f ␣共E兲f ␤共E⬘兲兴, 共8兲 where ␦共E − E⬘兲, ␦␣␤, ␦mn are Kronecker delta functions. In order to compute the average current along the channel and the power spectral density of the current fluctuations, we need to write the expectation values of the products of two and four operators,13 具a␣† m共E兲a␤n共E⬘兲典␴ = ␦共E − E⬘兲␦␣␤␦mn␴␣m共E兲,

共9兲

具a␣† m共E兲a␤n共E⬘兲a␥† k共E⬙兲a␦l共E⵮兲典␴ = ␦共E − E⵮兲␦共E⬘ − E⬙兲␦␣␦␦ml␦␤␥␦nk␴␣m共E兲关1 − ␴␥k共E⬙兲兴

2

e h

dE

共12兲 This expression consists of four terms, related to states at the source contacts, that can be evaluated by means of Eqs. 共9兲 and 共10兲, the first one 共F++兲 represents the correlation of fluctuations in two ingoing streams, the second and the third ones 共F+− , F−+兲 describe the correlations of the fluctuations of the ingoing and outgoing streams, the fourth one 共F−−兲 refers to two outgoing streams. The first term F++ can be expressed as

冉冊冕冕 e h

F++ =

=

e h e h

冕再兺冕再兺 dE

具关t†t兴nn␴Sn典s −

n苸S

dE

n苸S

兺具关t⬘†t⬘兴kk␴Dk典s

k苸D

F+− = −

具关t˜兴S;nn␴Sn典s −

兺具关t˜兴D;kk␴Dk典s

k苸D

冎

冎

, 共11兲

where 关t˜兴␣;lp ⬅ 关t†t兴lp if ␣ = S and 关t⬘†t⬘兴lp if ␣ = D 共l , p 苸 ␣兲. The unitarity of the matrix s has also been exploited, from which the relation r†r + t†t = 1 follows. It is easy to show that for a non-interacting channel, i.e., when occupancy of injected states does not affect transmission and reflection probabilities, Eq. 共11兲 reduces to the two-terminal Landauer’s formula.19 In general, we can observe that for an interacting channel Eq. 共11兲 provides a different result with respect to Landauer’s formula, because fluctuation of transmission probabilities induced by random injection in the device, is responsible for rectification of the current. The effect is often very small, but not always.20 However, it cannot be captured by Landauer’s formula, as other many-particle processes affecting device transport properties.21,22 The mean squared current reads

2

dE

兺

dE⬘

具␴Sm共E兲␴Sn共E⬘兲典s ,

共13兲

m,n苸S

2 since 具␴Sm 共E兲典s = 具␴Sm共E兲典s = f S共E兲 ∀m 苸 S. Correlations between ingoing states are established through the statistical expectation values of each couple of occupancies of states injected from the source. The second contribution F+− reads

共10兲

具I典 =

+ + 兵具NSm 共E兲NSn 共E⬘兲典兺 m,n苸S

− − + 具NSm 共E兲NSn 共E⬘兲典其 = F++ + F+− + F−+ + F−− .

+ ␦共E − E⬘兲␦共E⬙ − E⵮兲␦␣␤␦nm␦␥␦␦kl␴␣m共E兲␴␥k共E⬙兲,

where the first contribution in Eq. 共10兲 refers to exchange pairing 共␣ = ␦ , ␤ = ␥ , m = l , n = k兲, while the second to normal pairing 共␣ = ␤ , ␥ = ␦ , m = n , k = l兲.13 For the sake of simplicity, in the following we denote the expectation 具具典␴典s as 具典. By means of Eqs. 共4兲 and 共9兲 the average current reads

dE⬘

+ − − + 共E兲NSn 共E⬘兲典 − 具NSm 共E兲NSn 共E⬘兲典 − 具NSm

for ␣ ⫽ ␤ or m ⫽ n or E ⫽ E⬘. Including Eq. 共6兲 in Eq. 共7兲 and exploiting the relation ␴␣m共E兲2 = ␴␣m共E兲, the average of the product of two occupation numbers can be expressed as 具␴␣m共E兲␴␤n共E⬘兲典s

冉冊冕冕

冉冊冕冕再兺 2

e h

dE

dE⬘

具共1

m,l苸S

− 关t˜共E⬘兲兴S;ll兲␴Sm共E兲␴Sl共E⬘兲典s +

兺兺具关t˜共E⬘兲兴D;kk␴Sm共E兲␴Dk共E⬘兲典s

m苸S k苸D

冎

共14兲

,

since ␴␣2 l共E兲 = ␴␣l共E兲 ∀ l 苸 ␣ 共␣ = S , D兲, due to the Pauli exclusion principle. In Eq. 共14兲 correlations between ingoing and outgoing states are obtained by summing on each statistical average of the product of two occupation numbers of injected states, weighted with the reflection 共1 − 关t˜共E⬘兲兴S;ll = 关r†r共E⬘兲兴ll兲 or transmission probability 共关t˜共E⬘兲兴D;kk兲 of outgoing channels. By exploiting the anticommutation relations of the fermionic operators a, it is simple to demonstrate that the third term F−+ is identical to F+−. Indeed, − + + − 具NSm 共E兲NSn 共E⬘兲典 = 具NSn 共E⬘兲NSm 共E兲典.

共15兲

Finally, the fourth term F−− reads:

035329-3

F−− =

冉冊冕冉冊冕 e h

2

⌬E

兺兺具关t˜兴␣;ll共1 − 关t˜兴␣;ll兲␴␣l典s

␣=S,D l苸␣

2

e h

−

dE

⌬E

dE

兺兺

␣=S,D l,p苸␣

具关t˜兴␣;lp关t˜兴␣;pl␴␣l␴␣p典s

l⫽p

冉冊冕兺兺冓冋冕冉兺

−2

+

e h

e h

2

⌬E

dE

具关t⬘†r兴kp关r†t⬘兴 pk␴Dk␴Sp典s

k苸D p苸S

关t˜兴S;ll␴Sl −

dE

l苸S

兺

k苸D

冊册冔 2

关t˜兴D;kk␴Dk

s


BETTI, FIORI, AND IANNACCONE

+2

冉冊冕冕 2

e h

dE

兺 l苸S k苸D

dE⬘ 兺

⫻具关t˜共E⬘兲兴D;kk␴Sl共E兲␴Dk共E⬘兲典s

冉冊冕冕冉冊冕冕

+

2

e h

−2

dE

2

e h

dE⬘

dE⬘

dE

兺

具关t˜共E兲兴S;ll␴Sl共E兲␴Sp共E⬘兲典s

l,p苸S

兺

具␴Sl共E兲␴Sp共E⬘兲典s .

共16兲

l,p苸S

Equation 共16兲 contains all correlations between outgoing electron states in the source lead, where outgoing carriers at the source can be either reflected carriers incident from S or transmitted carriers injected from D. By means of the Eqs. 共13兲, 共14兲, and 共16兲, we find the mean squared current 具I2典 =

冉冊冕冉冊冕 e h

2

⌬E

␣=S,D l苸␣

2

e h

−


dE

⌬E

dE

兺兺 ␣=S,D

l,p苸␣

S共0兲 =


冉冊冕兺兺冓冋冕冉兺

+

e h

2

⌬E

dE

具关t⬘†r兴kp关r†t⬘兴 pk␴Dk␴Sp典s

k苸D p苸S

e h

关t˜兴S;ll␴Sl −

dE

l苸S

兺

冊册冔 2

关t˜兴D;kk␴Dk

k苸D

. s

共17兲

Finally, from Eqs. 共1兲, 共11兲, and 共17兲 the noise power spectrum can be expressed as S共0兲 =

冉冊冕冉冊冕 e2 ␲ប

e2 ␲ប

−

−2

+

dE

兺兺

␣=S,D l,p苸␣

冉冊冕 e2 ␲ប


␣=S,D l苸␣

dE

再冕

4␲ប e var ⌬E h


兺具关t⬘†r兴kp关r†t⬘兴pk␴Dk␴Sp典s

冉兺

n苸S

EFS

dE共Tr关t†t兴 − Tr关t†tt†t兴兲,

关t˜兴S;nn␴Sn −

兺关t˜兴D;kk␴Dk冊

k苸D

共19兲

EFD

冎

.

共18兲

Equation 共18兲 is the main theoretical result of this work, the power spectral density of the noise current is expressed in terms of transmission 共t , t⬘兲, reflection 共r兲 amplitude matrices, and properties of the leads, such as random occupation numbers of injected states. Let us point out that, although our derivation starts from Eq. 共2兲, which is valid only far from the mesoscopic interacting sample, Eq. 共18兲 allows to take into account both Pauli and Coulomb interactions through the dependence of t, t⬘ and r on actually injected states. Let us note that we go beyond the Hartree approximation by considering different random configuration of injected elec-

兺

ⴱ tⴱkltkptqp tql .

共20兲

k,q苸D

Each term of the sum can be interpreted as the coupling between a transmission event from channel p 苸 S into channel k 苸 D and from channel l 苸 S into channel q 苸 D, such a coupling is due to time-reversed transmissions from k into l and from q into p. In the same way, the third term of Eq. 共18兲 contains 关t⬘†r兴kp关r†t⬘兴 pk =

k苸D p苸S

dE

冕

关t†t兴lp关t†t兴 pl =

l⫽p

dE 兺

2e2 ␲ប

where EFS and EFD are the Fermi energies of the source and drain contacts, respectively. Such terms can be identified with partition noise 共PN兲 contribution. More in detail, the first term of Eq. 共18兲 is associated to the quantum uncertainty of whether an electron injected in the mode l from the reservoir ␣ is transmitted through or reflected by the barrier. The second term of Eq. 共18兲 contains instead 共l ⫽ p兲,

l⫽p

−2

tron states for different many-particle systems. There is a crucial difference with respect to LandauerBüttiker’s formula, since Eq. 共18兲 enables to consider fluctuations in time of the potential profile along the channel induced by the electrostatic repulsion between randomly injected electrons from the leads. Essentially, for each random configuration of injected states from both reservoirs, we consider a snapshot of device operation at a different time instant. All statistical properties—in the limit of zero frequency—can be obtained by considering a sufficient ensemble of snapshots. Let us discuss some physical limits of interest. First, we consider the case of zero temperature. In such condition the Fermi factor for populating electron states in the reservoirs is either 0 or 1, and all snapshots are identical, so the fourth term in Eq. 共18兲 disappears. In addition, we can remove the statistical averaging in Eq. 共18兲 and the first three terms lead to the following expression of the noise power spectrum,

ⴱ tⴱklrlprnp tkn , 兺 l,n苸S

共21兲

which represents the coupling between carriers transmitted from n 苸 S into k 苸 D and reflected from p 苸 S into l 苸 S. The second and third terms provide insights on exchange effects. Indeed, in such terms, contributions with k ⫽ q and l ⫽ n, respectively, are complex and they represent exchange interference effects 共fourth-order interference effects兲 in the many-particle wave function due to the quantum-mechanical impossibility to distinguish identical carriers.17 In the Sec. IV, we will be concerned with identical reservoirs, i.e., identical injected modes from the contacts. In this case the diagonal terms of the partition noise 关first term and part of the third term in Eq. 共18兲兴 will be referred as on-diagonal partition noise 共PN ON兲, while the off-diagonal ones 关second term and part of the third term in Eq. 共18兲兴 will be denoted as off-diagonal contribution to the partition noise 共PN OFF兲. Now let us assume that the number of quantum channel in the source is smaller than the one in the drain 共WS ⱕ WD兲 and let us consider the case of potential barrier wide with respect

035329-4



to the wavelength, so that one may neglect tunneling. In such a situation, the reflection amplitude matrix r is equal to zero for energies larger than the barrier maximum EC, whereas the transmission amplitude matrix is zero for energies smaller than EC. By means of the unitarity of the scattering matrix s, follows t†t = IS for E ⬎ EC, where IS is the identity matrix of order WS. Due to reversal time symmetry, there are WS completely opened quantum channels in the drain contact and WD − WS completely closed. In this situation only the fourth term in Eq. 共18兲 survives and the noise power spectral density becomes S共0兲 = =

2e2WS ␲ប

冕

+⬁

dE关f S共1 − f S兲 + f D共1 − f D兲兴

EC

2e2kTWS 关f S共EC兲 + f D共EC兲兴. ␲ប

共22兲

When EFS = EFD such term obviously reduces to the thermal noise spectrum 4kTG, where G = 关e2WS f S共EC兲兴 / 共␲ប兲 is the channel conductance at equilibrium. The fourth term in Eq. 共18兲 can be therefore identified with the injection noise 共IN兲 contribution. Equation 共18兲 describes correlations between transmitted states coming from the same reservoirs 关second term in Eq. 共18兲兴 and between transmitted and reflected states in the source lead 共third term兲, with a contribution of opposite sign with respect to the first term. The negative sign derives from Eq. 共10兲, in which exchange pairings include a minus sign due to the fermionic nature of electrons. Note that Eq. 共18兲 can be expressed in a symmetric form with respect to an exchange between the source and the drain contacts. Indeed, by exploiting the unitarity of the scattering matrix, the third term becomes,

冉冊冕冉冊冕

e2 − ␲ប 2

−

e ␲ប

dE 兺

具关t⬘†r兴kp关r†t⬘兴 pk␴Dk␴Sp典s 兺 k苸D p苸S

dE 兺

兺具关r⬘†t兴kp关t†r⬘兴pk␴Dk␴Sp典s , k苸D p苸S

2e2 ␲ប −

+

再冕

冕冕

dE

dE

dE

兺

␣=S,D

共Tr关t†t兴 − Tr关t†tt†t兴 + T␣兲f ␣

T␣ f ␣2 − 2 兺 ␣=S,D

兺

␣=S,D

冕

dE共Tr关t†t兴 − Tr关t†tt†t兴兲f S f D

冎

共Tr关t†tt†t兴 − T␣兲关f ␣共1 − f ␣兲兴 ,

S共0兲 =

共24兲

2e2 ␲ប

冕

dE兵关f S共1 − f S兲 + f D共1 − f D兲兴Tr关t†tt†t兴

+ 关f S共1 − f D兲 + f D共1 − f S兲兴共Tr关t†t兴 − Tr关t†tt†t兴兲其. 共25兲 Let us note that Eq. 共19兲 can be recovered as well from Eq. 共25兲. Indeed at zero temperature the stochastic injection vanishes since random statistics coincides to the Fermi factor. In the same way, Eq. 共22兲 might be derived from Eq. 共25兲, since in this case noise is only due to the thermionic emission contribution and fluctuations of the potential profile do not play any role in noise. III. COMPUTATIONAL METHODOLOGY AND QUANTITATIVE ANALYSIS

In order to properly include the effect of Coulomb interaction, we self-consistently solve the 3D Poisson equation, coupled with the Schrödinger equation with open boundary conditions, within the NEGF formalism, which has been implemented in our in-house open source simulator NANOTCAD VIDES.23 For what concerns the boundary conditions of Poisson equations, Dirichlet boundary conditions are imposed in correspondence of the metal gates, whereas null Neumann boundary conditions are applied on the ungated surfaces of the 3D simulation domain. In particular the 3D Poisson equation reads

ជ · 关⑀ⵜ ជ ␾共rជ兲兴 = − 关␳共rជ兲 + ␳ fix共rជ兲兴, ⵜ

共26兲

where ␾ is the electrostatic potential, ␳ fix is the fixed charge, which accounts for ionized impurities in the doped regions, and ␳ is the charge density per unit volume,

␳共rជ兲 = − e

共23兲

which establishes correlations between transmitted and reflected states in the source and drain leads. Now let us consider the limit when transmission and reflection matrices do not depend on random occupation numbers of injected states, i.e., a nonfluctuating potential profile is imposed along the channel. By exploiting the reversal time symmetry 共s = st, so that t⬘ = tt兲, the unitarity of the scattering matrix, Eq. 共18兲 reduces to Landauer-Büttiker’s noise formula,13 S共0兲 =

where T␣ = 兺l⫽p苸␣关t˜兴␣;lp关t˜兴␣;pl and the sum does not run on the spin. Equation 共24兲 then reduces to

冕冕

+⬁

dE

+e

Ei

−⬁

兺兺 DOS␣n共rជ,E兲␴␣n共E兲

␣=S,D n苸␣

Ei

dE

兺兺 DOS␣n共rជ,E兲关1 − ␴␣n共E兲兴,

␣=S,D n苸␣

共27兲 where Ei is the midgap potential, DOS␣n共rជ , E兲 is the local density of states associated to channel n injected from contact ␣ and rជ is the 3D spatial coordinate. From a computational point of view, modeling of the stochastic injection of electrons from the reservoirs has been performed by means of statistical simulations taking into account an ensemble of many electron states, i.e., an ensemble of random configurations of injected electron states, from both contacts. In particular, the whole energy range of integration 关Eqs. 共18兲 and 共27兲兴 has been uniformly discretized with energy step ⌬E. Then, in order to obtain a random injection configuration, a random number r uniformly distributed between 0 and 1 has been extracted for each electron state represented by energy E, reservoir ␣ and quantum channel n.24 More in detail, the state is occupied if r is

035329-5


BETTI, FIORI, AND IANNACCONE 4.4

10 metal gate

5

3.8 3.6 3.4 3.2

tox=1nm

0 nm

L=1

0

1nm

-5

d= 1n m

∆S(0) (%)

4

2

S(0) (pA /Hz)

4.2

1nm

tox=1nm

-4

∆E= 5x10 eV -3 ∆E= 10 eV -4 ∆E= 10 eV -5 ∆E= 5x10 eV LB's formula

3 100 200 300 400 500

Sample number a)

metal gate

-10 metal gate

-15 100 200 300 400 500

Sample number b)

tox=1nm 1nm

4nm

0 nm

L=1

1nm

4nm

FIG. 2. 共Color online兲 a兲 Noise power spectral density S共0兲 obtained from Eq. 共18兲 for a given potential as a function of current sample number for four different energy steps. b兲 Relative deviation of S共0兲 with respect to Landauer-Büttiker’s limit Eq. 共25兲. The simulated structure is the SNW-FET shown in Fig. 3.

smaller than the Fermi-Dirac factor, i.e., ␴Sn共E兲关␴Dn共E兲兴 is 1 if r ⬍ f S共E兲关f D共E兲兴, and 0 otherwise. The random injection configuration generated in this way has been then inserted in Eq. 共27兲 and self-consistent solution of Eqs. 共26兲 and 共27兲 and the Schrödinger equations has been performed. Once convergence has been reached, the transmission 共t , t⬘兲 and reflection 共r兲 matrices are computed. The procedure is repeated several times in order to gather data from a reasonable ensemble. In our case, we have verified that an ensemble of 500 random configurations represents a good trade-off between computational cost and accuracy. Finally, the power spectral density S共0兲 has been extracted by means of Eq. 共18兲. In the following, we will refer to self-consistent Monte Carlo simulations 共SC-MC兲, when statistical simulations using the procedure described above, i.e., inserting random occupations ␴Sn共E兲 and ␴Dn共E兲 in Eq. 共27兲, are performed. Instead, we will refer to SC simulations when the PoissonSchrödinger equations are solved considering f S and f D in Eq. 共27兲. SC-MC simulations of randomly injected electrons allow considering both the effect of Pauli and Coulomb interaction on noise. From a numerical point of view, particular attention has to be posed on the choice of the energy step ⌬E. In Fig. 2 the noise power spectrum computed by keeping fixed the potential profile along the channel and performing statistical Monte Carlo simulations of randomly injected electrons is shown for four energy steps. As already proved in Eq. 共24兲, the convergence to Landauer-Büttiker’s limit is ensured for all the considered energy steps: as can be seen, ⌬E = 5 ⫻ 10−4 eV provides faster convergence as compared to the other values with a relative error close to 0.16%. Let us point out that the NEGF formalism computes directly the total Green’s function G of the channel and the broadening function of the source 共⌫S兲 and drain 共⌫D兲 leads, rather than the scattering matrix s, that relates the outgoing waves amplitudes to the incoming waves amplitudes at different reservoirs. In order to obtain the matrix s, we have

tox=1nm metal gate

FIG. 3. 共Color online兲 3D structures and transversal cross sections of the simulated CNT 共top兲 and SNW-FETs 共bottom兲.

exploited the Fisher-Lee relation,25 which expresses the elements of the s-matrix in terms of the Green’s function G and transverse mode eigenfunctions 共see Appendix兲. IV. RESULTS

The approach described in the previous section has been used to study the behavior of shot noise in quasi-onedimensional 共1D兲 channel of CNT-FETs and SNW-FETs with identical reservoirs 共Fig. 3兲. We consider a 共13,0兲 CNT embedded in SiO2 with oxide thickness equal to 1 nm, an undoped channel of 10 nm and n-doped CNT extensions 10 nm long, with a molar fraction f = 5 ⫻ 10−3. The SNW-FET has an oxide thickness 共tox兲 equal to 1 nm and the channel length 共L兲 is 10 nm. The channel is undoped and the source and drain extensions 共10 nm long兲 are doped with ND = 1020 cm−3. The device cross section is 4 ⫻ 4 nm2. From a numerical point of view, a pz-orbital tight-binding Hamiltonian has been assumed for CNTs,26,27 whereas an effective mass approximation has been considered for SNWs28,29 by means of an adiabatic decoupling in a set of two-dimensional equations in the transversal plane and in a set of one-dimensional equations in the longitudinal direction for each 1D subband. For both devices, we have developed a quantum ballistic transport model with semi-infinite extensions at their ends. A mode space approach has been adopted, since only the lowest sub-bands take part to transport. In particular, we have verified that four modes are enough to compute the mean current both in the ohmic and saturation regions. All calculations have been performed at room temperature 共T = 300 K兲. Let us focus our attention on the Fano factor F, defined as the ratio of the actual noise power spectrum S共0兲 to the full shot noise 2q具I典. In Figs. 4 and 5 the contributions to F of

035329-6


STATISTICAL THEORY OF SHOT NOISE IN QUASI-… 1.1 1

1.1

-6

Full shot noise

0.9

CNT

10

1

-PN OFF

0.8

Fano factor

Fano factor

0.4 0.3

PN ON IN SC-MC LB

0

0.3

-10

0.2 0.1

-11

10 0

VGS-Vth (V) a)

0.2

-0.4 -0.2

0

0

-9

0.2

VGS-Vth (V) a)

VGS-Vth (V) b)

FIG. 4. 共Color online兲 Contributions to the Fano factor in a CNT-FET of the on-diagonal and off-diagonal partition noise and of the injection noise 关respectively, on-diagonal and off-diagonal part of the first three terms, and fourth term in Eq. 共18兲兴 as a function of the gate overdrive VGS − Vth for a drain-to-source bias VDS = 0.5 V. a兲 The on-diagonal partition 共PN ON, solid circles兲, the injection 共IN, open triangles up兲 and the full noise 共open circles兲 computed by means of SC-MC simulations are shown. The Fano factor computed by exploiting Landauer-Büttiker’s formula 共25兲 and SC simulations 共solid triangles down兲 is also shown. b兲 Off-diagonal partition noise contribution 共PN OFF兲 to F due to correlation between transmitted states and between transmitted and reflected states.

partition noise 关first three terms in Eq. 共18兲兴 and injection noise 共fourth term in Eq. 共18兲兲 are shown, as a function of the gate overdrive VGS − Vth for a drain-to-source bias VDS = 0.5 V for CNT-FETs and SNW-FETs, respectively, results have been obtained by means of SC-MC simulations. The threshold voltage Vth at VDS = 0.5 V is 0.43 V for the CNTFET and 0.13 V for the SNW-FET. In particular, Figs. 4共a兲 and 5共a兲 refer to the on-diagonal contribution to the partition noise 共solid circles兲, to the injection noise 共open triangles up兲 and to the complete Fano factor 共open circles兲 obtained by means of Eq. 共18兲, i.e., Pauli and Coulomb interactions simultaneously considered. We present also the Fano factor 共solid triangles down兲 computed by applying Eq. 共25兲 on the self-consistent potential profile, i.e., when only Pauli exclusion principle is included. In Figs. 4共b兲 and 5共b兲 we show the contribution of the off-diagonal partition noise to F, which provides a measure of mode-mixing and of exchange interference effects. As can be seen in Figs. 4共a兲 and 5共a兲, in the subthreshold regime 共VGS − Vth ⬍ −0.2 V , 具I典 ⬍ 10−9 A兲 the Poissonian noise for a nondegenerate injection is recovered, since electron-electron interactions are negligible due to the very small amount of mobile charge in the channel. In the strong inversion regime instead 共VGS − Vth ⬎ 0 V , 具I典 ⬎ 10−6 A兲, noise is greatly suppressed with respect to the full shot value. In particular for a SNW-FET, at VGS − Vth ⬇ 0.4 V 共具I典 ⬇ 2.4 ⫻ 10−5 A兲, combined Pauli and Coulomb interactions suppress shot noise down to 22% of the full shot noise value, while for CNT-FET the Fano factor is equal to 0.27 at VGS − Vth ⬇ 0.3 V 共具I典 ⬇ 1.4⫻ 10−5 A兲. This is due to the fact that as soon as an electron is injected, the barrier height

-PN OFF

-8

10

PN ON IN SC-MC LB

0 -0.4 -0.2

0.2

-7

10

0.4

10

-0.4 -0.2

SNW

0.5

-9

0.1

10

0.6

10

0.2

-6

0.7

-8

10

0.6 0.5

-5

10

0.9

-7

10

0.8 0.7

Full shot noise

0.4

10 -0.4 -0.2

0

0.2

0.4

VGS-Vth (V) b)

FIG. 5. 共Color online兲 Contributions to the Fano factor in a SNW-FET of the on-diagonal and off-diagonal partition noise and of the injection noise, obtained for VDS = 0.5 V, as a function of the gate overdrive VGS − Vth in a SNW-FET. In a兲 the on-diagonal partition, the injection and the full noise computed by means of SC-MC simulations 共both Pauli and Coulomb interactions taken into account兲 are shown together with results obtained by means of Eq. 共25兲. b兲 Off-diagonal partition noise due to correlation between transmitted states and between transmitted and reflected states.

along the channel increases, leading to a reduced transmission probability for other electrons. As shown in Fig. 4共a兲, the dominant noise source in ballistic CNT-FETs is the on-diagonal partition noise and the noise due to the intrinsic thermal agitations of charge carriers in the contacts 共injection noise兲, which is at most the 36% of the partition noise 共VGS − Vth ⬇ −0.1 V兲. Nearly identical results are shown for SNW-FETs, with the exception of a stronger contribution given by the injection noise, up to the 86% of the on-diagonal partition term 共VGS − Vth ⬇ −0.2 V兲. Moreover, the behavior of the two noise components, as a function of VGS − Vth, is very similar for both CNT- and SNW-FETs, F tends to 1 in the subthreshold regime, while in strong inversion regime shot noise is strongly suppressed. Let us stress that an SC-MC simulation exploiting Eq. 共18兲 is mandatory for a quantitative evaluation of noise. Indeed, by only considering Pauli exclusion principle through formula 共25兲, one would have overestimated shot noise by 180% for SNW-FET 共VGS − Vth ⬇ 0.4 V兲 and by 70% for CNT-FET 共VGS − Vth ⬇ 0.3 V兲.20,24 It is interesting to observe that the off-diagonal contribution to partition noise, due to exchange correlations between transmitted states and between transmitted and reflected states, has a strong dependence on the height of the potential profile along the channel 共variation of 5 orders of magnitude for CNT-FETs兲 and is negligible for quasi one-dimensional FETs. In particular, for CNT-FETs such term is at most 5 orders of magnitude smaller than the on-diagonal partition noise or injection noise in the strong inversion regime 共VGS − Vth ⬇ 0.3 V兲, while in the subthreshold regime its magnitude still reduces 共about 10−11 for VGS − Vth ⬇ −0.4 V兲. For SNW-FETs we have obtained similar results, the offdiagonal partition noise is indeed at most 5 orders of magnitude smaller than the other two contributions. In such conditions, transmission occurs only along separate quantum channels and an uncoupled mode approach is

035329-7


BETTI, FIORI, AND IANNACCONE 1.8

1 PN ON IN SC-MC LB

1.6 1.4

Fano factor

also accurate. Indeed, off-diagonal partition noise provides interesting information on the strength of the mode-coupling, which, as already seen, is very small. In particular, neglecting this term, results obtained from Eq. 共18兲 can be recovered as well. In the previous discussion, carriers from different quantum channels do not interfere. However, since we deal with a many indistinguishable particle system, such effects can come into play. To this purpose, we investigate in more detail two examples, in which exchange pairings, that include also exchange interference effects, give a non-negligible contribution to drain current noise. In the past exchange interference effects have been already predicted for example in ballistic conductor with an elastic scattering center in the channel,30 in diffusive four-terminal conductors of arbitrary shape31 and in quantum dot in the quantum Hall regime,32 connected to two leads via quantum point contacts. In the first case, we discuss, mode-mixing does not appear, i.e., the nondiagonal elements of the matrices t†t and t⬘†r are negligible with respect to the diagonal ones. Since the off-diagonal partition noise is negligible and since in the third term in Eq. 共18兲 only contributions with indices l = n = k = p survive, exchange interference effects do not contribute to electrical noise. We consider a CNT-FET at low bias condition: VDS = 50 mV. In Fig. 6共a兲 the on-diagonal partition noise, the injection noise and correlations due to the off-diagonal partition noise, evaluated performing statistical SC-MC simulations, are shown. In this case, on-diagonal correlations between transmitted and reflected states in the source lead 共in the same quantum channel兲 extremely affect noise. Indeed, at the energies at which reflection events in the source lead are allowed, also electrons coming from D can be transmitted into the injecting contact S, since the corresponding energy states in D are occupied and the barrier height is small. Instead the exchange correlations represented by the off-diagonal partition noise are negligible, since they are at least 5 order of magnitude smaller than the other three terms in Eq. 共18兲. Note that the noise enhancement obtained both in the inversion and subthreshold regimes is due to the fact that at low bias the current 具I典 becomes small, while the noise power spectrum S共0兲 tends to a finite value, because of the thermal noise contribution. Let now consider the situation in which modes are coupled and exchange interference effects, through the offdiagonal partition noise, contribute to drain current fluctuations. We consider the interesting case in which a vacancy, i.e., a missing carbon atom, is placed at the center of the channel of a 共13,0兲 CNT-FET. From a numerical point of view, this defect can be modeled by introducing a strong repulsive potential 共i.e., +8 eV, much larger than the energy gap of a 共13,0兲 CNT: Egap ⬇ 0.75 eV兲 in correspondence of such site, thus acting as a barrier for transmission in the middle of the channel 关Fig. 6共c兲兴. In Fig. 6共b兲 the three noise sources in Eq. 共18兲共on and off-diagonal partition noise, injection noise兲 are plotted as a function of the gate voltage VGS in the above threshold regime for VDS = 0.5 V, along with the full Fano factor computed performing SC and SC-MC simulations. Remarkably, in this case a mode space approach taking into account all modes 共i.e., 13兲 is mandatory in order to reproduce all cor-

0.8 0.7

1.2 1

-PN OFF

0.9

0.6

Full shot noise

0.5

0.8

0.4

0.6

0.3

0.4 0.2 0 -0.4

0.2 0.1

VDS= 0.05 V -0.2

0

0

0.2

VGS-Vth (V) a)

E (eV)

−0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 0

5

10

with a vacancy 0.5

0.6

0.7

0.8

VGS (V) b)

15

Z (nm)

20

25

1 0 30 2

−1

−2

X (nm)

c)

FIG. 6. 共Color online兲 a兲 Contributions to the Fano factor F by the on-diagonal partition noise 共solid circles兲, and the injection noise 共open triangles up兲 as a function of the gate overdrive VGS − Vth, for a drain-to-source bias VDS = 50 mV. The simulated device is a CNT-FET. The full noise computed by means of SC-MC simulations 共open circles, both Pauli and Coulomb interactions taken into account兲 and applying Eq. 共25兲共solid triangles down, only Pauli exclusion considered兲 is also shown. b兲 Contributions to F by the on-diagonal and off-diagonal partition noise and by the injection noise 关exploiting Eq. 共18兲兴 as a function of the gate overdrive for a CNT-FET with a vacancy in a site at the center of the channel. The drain-to-source bias is 0.5 V. c兲 Self-consistent midgap potential obtained by using the Fermi statistics for a gate voltage VGS = 0.7 V and a bias VDS = 0.5 V. Z is the transport direction along the channel, X is a transversal direction. The simulated device is the same of b兲.

relation effects on noise. As can be seen, off-diagonal exchange correlations gives rise to a not negligible correction to the Fano factor 共⬇4% of the full Fano factor at VGS = 0.8 V兲. We observe that such correlations are only established between transmitted electrons states 关second term in Eq. 共18兲兴, while correlations between reflected and transmitted electron states 关third term in Eq. 共18兲兴 are negligible since almost all electrons injected from the receiving contact D are reflected back because of the high bias condition. In this paper we have assumed phase-coherent quantum transport at room temperature. Our tools cannot include electronphonon interaction, that a room temperature may play a role even in nanoscale devices. Reference 33 has considered the effect of electron-phonon scattering and has neglected Coulomb interaction: they find that electron-phonon scattering

035329-8



increase shot noise in the above threshold regime, due to the broadening of the energy range of electron states contributing to transport. V. CONCLUSION

We have developed a novel and general approach to study shot noise in ballistic quasi one-dimensional CNT-FETs and SNW-FETs. By means of a statistical approach within the second quantization formalism, we have shown that the Landauer-Büttiker noise formula 关Eq. 共25兲兴 can be generalized to include also Coulomb repulsion among electrons. This point is crucial, since we have verified that by only using Landauer-Büttiker noise formula, i.e., considering only Pauli exclusion principle, one can overestimate shot noise by as much as 180%. From a computational point of view, we have quantitatively evaluated shot noise in CNT-FETs and SNW-FETs by self-consistently solving the electrostatics and the transport equations within the NEGF formalism, for a large ensemble of snapshots of device operation, each corresponding to a different configuration of the occupation of injected states. Furthermore, with our approach we are able to observe a rectification of the dc characteristics due to fluctuations of the channel potential, and to identify and evaluate quantitatively the different contributions to shot noise. We are also able to consider the exchange interference effects, which are often negligible but can be measurable when a defect, introducing significant mode mixing, is inserted in the channel. ACKNOWLEDGMENTS

lattice constant along the x direction. In the k-representation, for a conductor of uniform cross-section, we can exploit a mode representation in the transverse direction and a plane wave representation in the longitudinal direction and Eq. 共A.1兲 becomes snm = − ␦nm +

⌫S =

Let us consider a 2D channel of length L and denote with x and y the longitudinal direction and the transverse one, respectively. If the interface between the lead S 共D兲 and the conductor is defined by xS = 0共xD = 0兲, GDS共y D ; y S兲 = GDS共xD = 0 , y D ; xS = 0 , y S兲 represents the wave function at 共xD = 0 , y D兲 due to an excitation at 共xS = 0 , y S兲. In real space the Fisher-Lee relation reads,

冕冕 dy D

␹m共y S兲,

冉

⌫S;11 0 0 0

冊

⌫D = 2N⫻2N

冉

0 0 0 ⌫D;22

冊

, 2N⫻2N

បv共kSn兲

where ⌫S;11共n , m兲 = ␦nm L ∀ n , m 苸 S and ⌫D;22共n , m兲 បv共kD n兲 = ␦nm L ∀ n , m 苸 D. Generalization to a CNT-FET structure is straightforward. Let us indicate with NC and N M the number of carbon atoms rings and the number of modes propagating along the channel, respectively. Since the coupling between the identical reservoirs and the channel involve only the end rings of the channel, ⌫S and ⌫D are 共NM NC兲 ⫻ 共N M NC兲 diagonal matrix and the only nonzero blocks are the first one and the latter one, respectively: ⌫S;11共n,m兲 = ␦nm

បv共kn兲 L

⌫D;NcNc共n,m兲 = ␦nm

បv共kn兲 L

∀ n,m = 1, ¯ ,NM ,

∀ n,m = 1, ¯ ,NM . 共A.3兲

By exploiting Eqs. 共A.2兲 and 共A.3兲 we can find the transmission 共t兲 and reflection 共r兲 amplitude matrix,

APPENDIX

iប冑vnvm a

共A.2兲

where GDS共n , m兲 = GDS共n , kn ; m , km兲 and kn is the longitudinal wave vector of the transverse mode n. Let us assume both leads to be identical and denote with D 兵kS1 , . . . , kNS其共兵kD 1 , . . . , kN 其兲 the set of wave vectors associated to the N modes coming from the lead S 共D兲. Since the only nonzero components of the self-energy involve the endpoints, in the k-representation ⌫S and ⌫D can be expressed as

The work was supported in part by the EC Seventh Framework Program under the Network of Excellence NANOSIL 共Contract No. 216171兲, and by the European Science Foundation EUROCORES Program Fundamentals of Nanoelectronics, through funds from CNR and the EC Sixth Framework Program, under project DEWINT 共Contract No. ERAS-CT-2003-980409兲. The authors would like to thank M. Büttiker for fruitful discussion.

snm = − ␦nm +

iប冑vnvm GDS共n,m兲, L

dy S␹n共y D兲GDS共y D ;y S兲共A.1兲

where n is a mode outgoing at lead D with velocity vn, m is a mode incoming at lead S with velocity vm and a is the

tnm = i冑⌫D;NCNC共n,n兲GNC1共n,m兲冑⌫S;11共m,m兲, rnm = − ␦nm + i冑⌫S;11共n,n兲G11共n,m兲冑⌫S;11共m,m兲. 共A.4兲 Since at zero magnetic field t⬘ = tt, relations Eq. 共A.4兲 is all we need to compute the power spectral density Eq. 共1兲 from Eq. 共18兲. A similar procedure has been adopted for SNWFETs where, from a computational point of view, the channel has been discretized in a sequence of slices in the longitudinal direction. In this case equations in Eq. 共A.4兲 are obtained as well, but replacing the number of rings with the number of slices.

035329-9


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